Well sounds to me like your exhaust piping is not insulated. What happens is the warm air from your bathroom condenses inside the pipe and runs back into the fan box then into your bathroom. Its because of the piping being exposed to cold air in your attic or wall wherever it might be run through.

An easy fix you can do yourself is wrap the exhaust pipe in your attic with pipe insulation available at most hardware stores like Home Depot etc.

I'm sure there must be someone in your vicinity that can help you out if you can't or don't want to do it yourself.

Within hours you'll be comparing offers from top-rated professionals. It's free to post and you're under no obligation to hire.

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